Indefinite Integrals of Logarithmic Functions

# Indefinite Integrals of Logarithmic Functions

Theorem 1: If $f(x) = \log x$, then $\int \log x = x\log x - x + C$. If $f(x) = \log_a x$, then $\int \log_a x = \frac{x\log x - x}{\log a } + C$. |

The property above now allows us to integrate logarithmic functions. Let's now look at some examples.

# Example 1

**Evaluate the integral $\int \log_3 x + \cos x$.**

Applying the property above and what we already know about integrating trigonometric functions, we get that:

(1)\begin{align} \int \log_3 x + \cos x = \frac{x\log x - x}{\log 3} + \sin x \end{align}

# Example 2

**Evaluate the integral $\int \log_7 x^2 - \log_7 x \: dx$.**

Applying logarithmic rules, we can simplify this drastically to get:

(2)\begin{align} \int \log_7 2x^2 - \log_7 x \: dx = \int \log_7 \left ( \frac{x^2}{x} \right ) \: dx \\ \int \log_7 2x^2 - \log_7 x \: dx = \int \log_7 \left ( x \right ) \: dx \\ \int \log_7 2x^2 - \log_7 x \: dx = \frac{x \log x - x}{\ln 7} + C \end{align}